Finite Group Databasebeta

← #30605291

Subgroups — #30605291

Order 15616.

432 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21461720C211yesmaximal, characteristic, complemented
2#21461052C211yesmaximal, characteristic, complemented
3#21460818C212yesmaximal, complemented
4#21460801C212yesmaximal, complemented
5#21451379C211yesmaximal, characteristic
6#16840778—21no—
7#16840734—21no—
8#16840642—21no—
9#16840642C2212yes—
10#16840608C2212yes—
11#16840606—21no—
12C2×#14519734C2211yescharacteristic
13#16839700C2211yescharacteristic
14#16839697C411yescharacteristic
15#16839676C2211yescharacteristic
16#16839670C411yescharacteristic
17#16839650—21no—
18#16839641—21no—
19#16839638—21no—
20#16839635—21no—
21#16839630C411yescharacteristic
22C2×C4×D4×C61C411yescharacteristic
23C2×#11758217—22no—
24C4⋊#92420—41no—
25#14519834—41no—
26#14519834—41no—
27C22⋊#92420—22no—
28C4×#92420—41no—
29C22×C22⋊D61—21no—
30C22×C4×D61D411yescharacteristic
31#14519734—41no—
32#14519734—41no—
33C2×#11758196—21no—
34C2×#11758196D411yescharacteristic
35C2×#11758196C2311yescharacteristic
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—21no—
39C2×#11758198—21no—
40C2×#11758198C2×C411yescharacteristic
41C2×#11758198D411yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684C2×C411yescharacteristic
48#14519684C2×C411yescharacteristic
49C22×D4×C61C2×C411yescharacteristic
50C23×C4×C61—21no—
51C4×D4×C61—41no—
52C4×D4×C61—41no—
53C2×C4⋊C4×C61C2×C411yescharacteristic
54C2×C22⋊C4×C61—21no—
55C2×C42×C61C2×C411yescharacteristic
56C24×C61—21no—
57C23×D61C22⋊C411yescharacteristic
58C2×D4×C61—41no—
59C2×D4×C61—42no—
60C22×C4×C61—21no—
61C22×C4×C61—42no—
62C22×C4×C61—81no—
63C22×C4×C61—81no—
64C22×C4×C61C22⋊C411yescharacteristic
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22×C411yescharacteristic
67C2×C22⋊D61—42no—
68C2×C22⋊D61—42no—
69C2×C22⋊D61—81no—
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C2×D411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C2×C4×D61—21no—
76C2×C4×D61—21no—
77C2×C4×D61—22no—
78C2×C4×D61—41no—
79#11758217—42no—
80#11758217—42no—
81C2×#92420—42no—
82C2×#92420—42no—
83C4⋊C4×C61—41no—
84C22⋊C4×C61—41no—
85C22⋊C4×C61—41no—
86C42×C61—21no—
87C42×C61—21no—
88C22⋊Dic61—42no—
89C22⋊Dic61—42no—
90C22⋊Dic61—81no—
91#11758198—41no—
92#11758198—41no—
93#11758198—81no—
94#11758196—41no—
95#11758196—41no—
96#11758196—81no—
97C4×Dic61—21no—
98C4×Dic61—21no—
99C23×C61—22no—
100C23×C61—41no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61—41no—
104C23×C61—41no—
105C23×C61—41no—
106C23×C61C2×C22⋊C411yescharacteristic
107C22×D61—21no—
108C22×D61—21no—
109C22×D61—21no—
110C22×D61—41no—
111C22×D61—41no—
112D4×C61—42no—
113D4×C61—42no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—84no—
124C2×C4×C61#9911yescharacteristic
125C2×C4×C61#9911yescharacteristic
126C22⋊D61—84no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—21no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61#13512yes—
140C4×D61—42no—
141C4×D61—42no—
142#92420—82no—
143#92420—82no—
144#11847—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—21no—
147C22×C61—42no—
148C22×C61—42no—
149C22×C61—42no—
150C22×C61—81no—
151C22×C61—81no—
152C22×C61—81no—
153C22×C61—81no—
154C22×C61C2×#9911yescharacteristic
155C22×C61#41611yescharacteristic
156C22×C61#42111yescharacteristic
157C2×D61—42no—
158C2×D61—41no—
159C2×D61—41no—
160C2×D61—41no—
161C2×D61—81no—
162C4×C61—21no—
163C4×C61—21no—
164C4×C61—41no—
165C4×C61—41no—
166C4×C61—81no—
167Dic61—22no—
168Dic61—41no—
169Dic61—41no—
170Dic61—81no—
171Dic61—81no—
172#2598—611no—
173#1822—611no—
174#1458—611no—
175#1353—612no—
176#1345—612no—
177C2×C61—21no—
178C2×C61—21no—
179C2×C61—82no—
180C2×C61#151611yescharacteristic
181C2×C61#212011yescharacteristic
182C2×C61#213411yescharacteristic
183D61—81no—
184C2×C4×D4#692311yescharacteristic
185C2×#117—611no—
186#431—1221no—
187#407—1221no—
188#400—611no—
189#399—611no—
190#394—611no—
191#392—611no—
192#388—1221no—
193#386—1221no—
194#386—1221no—
195#385—1221no—
196#383—611no—
197#343—612no—
198#343—1221no—
199#324—612no—
200#323—1221no—
201C61#1184711yescharacteristic, complemented, Sylow
202C22×D4—1221no—
203C22×D4C2×#692311yescharacteristic
204C23×C4—21no—
205C23×C4—611no—
206C2×M16—1222no—
207C4×D4—41no—
208C4×D4—41no—
209#117—2441no—
210#117—2441no—
211C2×C4⋊C4—611no—
212C2×C4⋊C4—611no—
213C2×C4⋊C4—1221no—
214C2×C4⋊C4C2×#692311yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—611no—
217C2×C22⋊C4—611no—
218C2×C22⋊C4—1221no—
219C2×C22⋊C4—1221no—
220C2×C22⋊C4—1221no—
221C2×C22⋊C4—1221no—
222C2×C42—611no—
223C2×C42C2×#692311yescharacteristic
224C4⋊C8—2441no—
225C22⋊C8—1222no—
226C22⋊C8—2441no—
227C22⋊C8—2441no—
228C4×C8—2441no—
229#95—611no—
230#95—611no—
231#95—1221no—
232#95—1221no—
233#95—1221no—
234#95—1221no—
235C24—21no—
236C24—611no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—4881no—
240C2×D4—2442no—
241C2×D4—2442no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—611no—
247C22×C4—611no—
248C22×C4—611no—
249C22×C4—1222no—
250C22×C4—1221no—
251C22×C4—1221no—
252C22×C4—1221no—
253C22×C4—1221no—
254C22×C4—2441no—
255C22×C4C22⋊#692311yescharacteristic
256C22×C4C22⋊#692311yescharacteristic
257C22×C4C22×#692311yescharacteristic
258M16—2442no—
259M16—2442no—
260C2×C8—2442no—
261C2×C8—2442no—
262C4⋊C4—41no—
263C4⋊C4—4881no—
264C4⋊C4—2441no—
265C4⋊C4—2441no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—4881no—
269C22⋊C4—4881no—
270C22⋊C4—2441no—
271C22⋊C4—2441no—
272C22⋊C4—2442no—
273C22⋊C4—2442no—
274C42—21no—
275C42—21no—
276C42—1221no—
277C42—1221no—
278C23—22no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—1221no—
286C23—1221no—
287C23—1221no—
288C23—2441no—
289C23—2441no—
290C23C2×C22⋊#692311yescharacteristic
291D4—42no—
292D4—42no—
293D4—4884no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—84no—
304C2×C4—612no—
305C2×C4—1221no—
306C2×C4—1221no—
307C2×C4—1221no—
308C2×C4—1221no—
309C2×C4—2442no—
310C2×C4—2442no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4—2441no—
316C2×C4—2441no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4#1451983511yescharacteristic
320C2×C4#1451983511yescharacteristic
321C8—4882no—
322C8—4882no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—4881no—
333C22—2442no—
334C22—2441no—
335C22—2441no—
336C22—2441no—
337C22C2×#1451983511yescharacteristic
338C22#1684075211yescharacteristic
339C22#1684075911yescharacteristic
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—81no—
345C4—1222no—
346C4—4881no—
347C4—4881no—
348C4—2441no—
349C4—2441no—
350C2—21no—
351C2—21no—
352C2—82no—
353C2—4881no—
354C2#2146124511yescharacteristic
355C2#2146207011yescharacteristic
356C2#2146211811yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×D4×C61C4
socleC22×C61C2×#99
prime radical#16839670C4
cosocleC2×#11758196C23
hypercenterC2×C4×D4#6923
hypocenterC61#11847
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #21461720C2semidirect
2. #21461052C2semidirect
3. #21460818C2semidirect
4. #21460801C2semidirect
201. C61#11847semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214617202
#214610522
#214608182
#214608012
#214513792
#1184761